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Venn Diagrams [from Google Images] If you are a really cool mathematician you get to have your name on a theorem like Rolle, or have your name on a triangle like Pascal, or even your name on a diagram like John Venn. From Mental Floss From Snorg Tees From Google Images From Google Images: As you can probably guess, IB loves Venn diagrams! So we should become experts. All Venn diagrams have the same set-up: A rectangle which can represent a sample space Circle(s) which can represent events This picture is from our textbook The complement of A, indicated by A’, represents the non-occurrence of event A. So the Venn diagram can look like: Since we can represent a sample space with a set, then set notation is used with Venn diagrams. x ∈ A is read as “x is an element of set A” n (A) is read as “the number of elements in set A” When there is more than one circle in the Venn diagram, then we can be concerned with the union of the sets and the intersection of the sets. I can remember the “union” starts with the letter “u” and the union symbol looks like a “u” ∪ [And it is a good thing to be united, so we should smile!] I don’t have a clever way to remember the “intersection” symbol. Sorry! ∩ If two sets have no elements in common, then they are considered to be disjoint sets. This means that A ∩ B = Ø Note: Ø is the symbol for the empty set so don’t use it if you mean “zero”. If A ∩ B = Ø, then A and B are said to be “mutually exclusive”. Note: Our textbook says that it is impossible to represent independent events on a Venn diagram. We can have more than two circles. Here is Wikipedia’s look at Venn diagrams with more than two events: An example from our textbook: One way to figure out how to fill in the Venn Diagram: 50 married men were asked whether they gave their wife flowers or chocolates for their last birthday. [I got a pink Blackberry.] The results were: 31 gave chocolates, 12 gave flowers, and 5 gave both chocolates and flowers. If one of the married men was chosen at random, determine the probability that he gave his wife: (a) chocolates or flowers (b) chocolates but not flowers (c) neither chocolates nor flowers (d) flowers if it is known that he did not give her chocolates First construct a Venn Diagram to illustrate the problem. Let = number who gave chocolates Let = number who gave flowers Another example: Suppose S = x : x < 100 and x ∈ Z + { } Let A = multiples of 7 in S And B = multiples of 5 in S Find the number of elements in the following (a) A (b) B (c) A ∩ B (d) A ∪ B Can we find a shortcut to find (d)? [Use parts, a, b, and c]